> For the complete documentation index, see [llms.txt](https://docs.1l.lol/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.1l.lol/number-theory/sieve-eratosthenes.md).

# 埃氏筛法

### 思想

埃氏筛（埃拉托斯特尼筛）从 2 开始，把每个素数的倍数都标记为合数，剩下的就是素数。复杂度约 **O(n log log n)**。

### 实现

```cpp
const int N = 1000000;
bool isPrime[N + 1];

void sieve(int n) {
    fill(isPrime, isPrime + n + 1, true);
    isPrime[0] = isPrime[1] = false;
    for (int i = 2; i * i <= n; i++)
        if (isPrime[i])
            for (int j = i * i; j <= n; j += i)   // 从 i*i 开始
                isPrime[j] = false;
}
```

### 关键点

* 内层从 `i * i` 开始，因为小于它的倍数已被更小的素因子筛过。
* 时间复杂度 O(n log log n)，空间 O(n)，适合 n 到 10⁷ 左右。

### 求 2\~n 的所有素数

```cpp
sieve(n);
for (int i = 2; i <= n; i++)
    if (isPrime[i]) cout << i << " ";
```

### 缺点

同一个合数会被多个素因子重复标记（如 12 被 2、3 标记两次）。**线性筛法**可以做到每个合数只标记一次，复杂度 O(n)。


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